Evaluate the Integral

Problem

(∫_^)(1/√(,1−9*x2)*d(x))

Solution

1. Identify the form of the integral as a standard inverse trigonometric integral, specifically (∫_^)(1/√(,1−u2)*d(u))=arcsin(u)+C

2. Rewrite the term 9*x2 as a perfect square (3*x)2 to match the standard form.

3. Apply the substitution u=3*x which implies d(u)=3*d(x) or d(x)=1/3*d(u)

4. Substitute these values into the integral to get (∫_^)(1/√(,1−u2)1/3*d(u))

5. Factor out the constant 1/3 and integrate to obtain 1/3*arcsin(u)+C

6. Back-substitute u=3*x to find the final result in terms of x

Final Answer

(∫_^)(1/√(,1−9*x2)*d(x))=1/3*arcsin(3*x)+C

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